Thickness and a gap lemma in Rd\mathbb{R}^d

We give a definition of thickness in $\mathbb{R}^d$ that is useful even for totally disconnected sets, and prove a Gap Lemma type result. We also guarantee an interval of distances in any direction in thick compact sets, relate thick sets (for this definition of thickness) with winning sets, give a lower bound for the Hausdorff dimension of the intersection of countably many of them, a result guaranteeing the presence of large patterns, and lower bounds for the Hausdorff dimension of a set in relationship with its thickness.Comment: 19 page

On the volumes of simplices determined by a subset of Rd\mathbb{R}^d

We prove that for $1\le k<d$, if $E$ is a Borel subset of $\mathbb{R}^d$ of Hausdorff dimension strictly larger than $k$, the set of $(k+1)$-volumes determined by $k+2$ points in $E$ has positive one-dimensional Lebesgue measure. In the case $k=d-1$, we obtain an essentially sharp lower bound on the dimension of the set of tuples in $E$ generating a given volume. We also establish a finer version of the classical slicing theorem of Marstrand-Mattila in terms of dimension functions, and use it to extend our results to sets of ``dimension logarithmically larger than $k$''.Comment: 12 pages, no figure

On the volumes of simplices determined by a subset of Rd

We prove that for 1 ≤ k < d, if E is a Borel subset of Rd of Hausdorff dimension strictly larger than k, the set of (k+1)-volumes determined by k+2 points in E has positive one-dimensional Lebesgue measure. In the case k = d−1, we obtain an essentially sharp lower bound on the dimension of the set of tuples in E generating a given volume. We also establish a finer version of the classical slicing theorem of Marstrand-Mattila in terms of dimension functions, and use it to extend our results to sets of “dimension logarithmically larger than k”

Intersections of thick compact sets in ℝd

Funding: Alexia Yavicoli was financially supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.We introduce a definition of thickness in ℝd and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove that given any compact set in ℝd with thickness τ, there is a number N(τ) such that the set contains a translate of all sufficiently small similar copies of every set in ℝd with at most N(τ) elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.Publisher PDFPeer reviewe

Fractals, patterns and dimension

Es bien sabido que si un conjunto tiene medida Lebesgue positiva, entonces contiene una copia homotética de cualquier conjunto finito. Surge entonces la pregunta natural: ¿Cuán chico puede ser un conjunto que contenga muchas configuraciones geométricas? En esta tesis demostraré entre otros resultados, que existe un conjunto chico y cerrado (definido explicitamente), sin puntos aislados, que contiene todo patrón finito definido por una familia de funciones que cumple ciertas condiciones. Entre otras aplicaciones, veremos que hay un conjunto de dimensión de Hausdorff cero que contiene todo patrón polinomial finito (en una o varias variables). También veremos que el conjunto de funciones bilipschitz satisfacen las condiciones, lo cual generaliza resultados anteriores sobre funciones lineales.Uno puede hacerse la pregunta en cierto sentido opuesta: ¿Cuán grande puede ser un conjunto que no contenga ciertos patrones? En esta tesis respondo la pregunta en el caso de patrones lineales. Veremos que dados contables patrones lineales, existe un conjunto compacto (definido explicitamente) que no contiene ninguno de esos patrones y tiene dimensión de Hausdorff total, y más aún tiene medida de Hausdorff positiva para cualquier función de dimensión prefijada.Los resultados anteriores muestran que si consideramos como noción de tamaño a la dimensión de Hausdorff, hay conjuntos grandes sin ciertos patrones, como asi también conjuntos chicos con muchos patrones. Otra noción de tamaño importante es el espesor, definido por Newhouse. En esta tesis desarrollaré un trabajo en el que muestro que si un conjuntode Cantor tiene espesor grande entonces contiene progresiones aritméticas largas, como asi también patrones más generales. Además mostraré un resultado en el que estudio el tamaño (dimensiones Lq) de las proyecciones de una clase de medidas autosimilares aleatorias. En el momento de la publicación de este trabajo no se sabia casi nada para la dimensión Lq de medidas fractales con estructurade solapamiento.It is well known that if a set has positive Lebesgue measure, then it contains a homothetic copy of any finite set. The natural question then arises: How small can be a set that contains many geometrical configurations? In this thesis I will prove among other results, that there exists a small and closed set (explicitly defined), without isolated points, containing all finite patterns defined by a family of functions satisfying certain conditions . Among other applications, we will see that there exists a set of Hausdorff dimension zero that contains all finite polynomial patterns (in one or more variables). We will also see that the set of bilipschitz functions satisfies the conditions, which generalizes previous results on linear functions. One can ask what is in some sense the opposite question: How large can be a set that does not contain certain patterns? In this thesis I answer the question in the case of linear patterns. We will see that given countably many linear patterns, there is a compact set (explicitly defined) that does not contain any of those patterns and has full Hausdorff dimension, and even more, has positive Hausdorff measure for any given dimension function. The previous results show that if we consider the Hausdorff dimension as a notion of size, there are large sets without certain patterns, as well as small sets with many patterns. Another important notion of size is thickness, defined by Newhouse. In this thesis I will develop a work in which I show that if a Cantor set has large thickness then it contains long arithmetic progressions, as well as more general patterns. In addition, I will develop a result in which I study the size (L q dimensions) of the projections of a class of random self-similar measures. At the time of its publication, almost nothing was known for the L q dimension of fractal measures with an overlapping structure.Fil: Yavicoli, Alexia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

The density of sets containing large similar copies of finite sets

Funding: VK is supported by the Croatian Science Foundation, project n◦ UIP-2017-05-4129 (MUNHANAP). AY is supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.We prove that if E⊆Rd (d≥2) is a Lebesgue-measurable set with density larger than n−2n−1, then E contains similar copies of every n-point set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point sets tends to 1 at a rate 1−O(n−1/5log n).PreprintPeer reviewe